Izzo told reporters late Thursday night that "it won't go on forever, I can tell you that."
The coach had just arrived back in East Lansing, Mich., after traveling to Cleveland on Thursday afternoon with his wife, daughter and son.
Izzo said: "I feel bad that I can't talk. I feel good that I did what I had to do."
Izzo is apparently closer to saying yes to the Cavaliers than previously thought, a source told ESPN.com's Andy Katz on Thursday.
That source discussed Izzo's future with a close confidant of Izzo's who is coaching in the NBA. The source told Katz that Izzo, who talked to his team earlier this week about his interest in the Cavaliers, is leaning toward taking the job if it is offered.
"Three weeks ago that wasn't the case, but it has changed,'' the source said.
After flying to Cleveland in one of Cavs owner Dan Gilbert's jets, Izzo spent several hours visiting with members of the front office and touring the team's facilities on Thursday.
Izzo, who has been courted by pro teams in the past, is believed to be mulling a contract worth up to $6 million a season from the Cavaliers, who fired Mike Brown last month after they lost in the second round of the playoffs to Boston despite having the league's best regular-season record for the second straight year.
It is not known if Izzo gave the Cavs a decision or a timetable for one.
New Cleveland general manager Chris Grant said the Cavaliers have spoken to a "number" of candidates, but did not divulge any names. The team has inquired about Milwaukee assistant Kelvin Sampson.
Izzo landed at County Airport in Richmond Heights, Ohio. His plane, which left East Lansing, Mich., was diverted there because of the number of TV cameras at Burke Lakefront Airport, the person said. Gilbert did arrive at Burke, which is a short drive from Quicken Loans Arena, the Cavs' downtown home.
"Coach Izzo made me aware he is meeting with Cleveland," Michigan State athletic director Mark Hollis said in a statement. "The details of any meetings are between him and me."
Gilbert has been ultra-secretive in his pursuit of Izzo, whom he has known for years.
Grant, who recently took over when Danny Ferry resigned, confirmed this week the team has had contact with Izzo but would not comment on whether an offer has been made to the 55-year-old coach. Gilbert is a Michigan State graduate.
Izzo isn't the only coach interested in the Cavaliers job, which may or may not include the chance to coach free agent LeBron James. Wednesday night, an NBA Eastern Conference executive with knowledge of the Cavaliers' thinking told ESPN The Magazine's Chris Broussard that Byron Scott is a strong candidate to be offered the job.
The executive told Broussard that Scott is on equal footing with Izzo.
Scott, a former Los Angeles Lakers guard who has coached the New Jersey Nets and the New Orleans Hornets, has spoken with members of the Cavaliers' front office the past few days. Late Wednesday night the talks took on a more serious nature, the source told Broussard.
AOL Fanhouse.com reported that Scott had an hour-long interview with Grant and assistant GM Lance Blanks.
Izzo, meanwhile, said recently he wouldn't leave Michigan State until the school won another national championship. But that feeling apparently has changed. He met with his players Tuesday to update them on the Cavs job, according to Spartans associate head coach Mark Montgomery.
Izzo did not give any indication of whether he would take the job, but a source told Katz that Izzo told the team he has a lot of thinking to do.
A member of the Michigan State Board of Trustees said earlier this week that Izzo already has turned down an offer from the Chicago Bulls this offseason.
A source within the Bulls' organization denied that to ESPNChicago.com's Nick Friedell, though the source did say Izzo and the Bulls had talked.
About 500 people, including center Derrick Nix, showed up at the Magic Johnson statue outside Michigan State's arena, responding to a rally created by fans on message boards and social networking websites.
"Coach is the best thing that ever happened to me," said Nix, who was a freshman last season.
Hundreds wrote personal messages in different colors on a huge banner that was taped up to the windows of Izzo's office. An "Oh no, please don't go Izzo," banner was taped at the base of Johnson's statue.
Izzo has been at the school since 1983 and has been the Spartans' coach since 1995, leading them to six Final Fours in the past 12 years. He has spurned previous overtures from the NBA, most notably from the Atlanta Hawks, who offered him a five-year deal in 2000.
Izzo must weigh leaving a familiar situation that pays $3 million a year for probably as long as he wants the job, and perhaps a legacy that would put him among college basketball's all-time greats, for a chance to coach in a player-first league and a team that doesn't know if it will have James next season.
In recent years, several successful college coaches, including Rick Pitino, Tim Floyd and John Calipari, have struggled in switching to the pro game. Several of Izzo's former players believe he can make the leap.
"I don't think coach would have a problem getting respect in an NBA locker room," New Orleans Hornets guard Morris Peterson said. "I think guys will buy into what he's telling them because he has a gift with people. Izz is one of the smartest coaches in basketball and one of the nicest guys around."
Milwaukee guard Charlie Bell said Izzo may have to work on his intensity.
"I've heard he's not as hard on guys as he used to be," Bell said. "He'd have to tone it down even more in the pros."
Information from ESPN.com college basketball reporter Andy Katz, ESPN The Magazine's Chris Broussard, ESPNChicago.com's Nick Friedell and The Associated Press was used in this report.